Re: How to calculate the conductor of an abelian extension such as Q[x]/(x^3- 19x -19)

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Mon, Nov 11, 2024 at 08:51:22AM +0100, Karim Belabas wrote:
> * David Bernier [2024-11-11 05:08]:
> > I'm interested in cubic extensions of Q that are abelian, in connection with
> > a probable prime test. I have a list of cubic polynomials f_1, ... f_22 and
> > I want to find the first f_i such that f_i is irreducible over F_p, where p
> > can be assumed prime. For a given f_i, I noticed a periodicity in p of the
> > irreducibility character of f_i over F_p (ref. Mathematics Stack Exchange at
> > the link: https://math.stackexchange.com/questions/4995484/irreducibility-of-cubic-polynomials-over-finite-fields-f-p
> > ). User leoli1 mentioned as relevant the conductor N of the splitting field
> > of f_i. I have f_8 = X^3 - 19X - 19 with discriminant 133^2.
> 
> (You meant 19^2.)

More precisely, the discriminant of the polynomial is 133^2,
the discriminant of the field is 19^2.

rnfconductor gives you the prime decomposition law:

If P is of conductor f, rnfconductor will gives you a subgroup H of (Z/fZ)^*
such that for all p not dividing the discriminant of P,
P factors mod p if and only if (p mod f) belongs to H.

Cheers,
Bill.